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1: 28.17 Stability as $x\to\pm\infty$
§28.17 Stability as $x\to\pm\infty$
For real $a$ and $q$ $(\neq 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves $a=a_{n}\left(q\right)$ and $a=b_{n}\left(q\right)$; compare Figure 28.2.1. …
2: 12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U\left(a,b,x\right)$ and $M\left(a,b,x\right)$13.2(i)) whose regions of validity include intervals with endpoints $x=\infty$ and $x=0$, respectively. …
3: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point $z_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\tfrac{1}{3}$9.6(i)). These expansions are uniform with respect to $z$, including the turning point $z_{0}$ and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … In regions in which the function $f(z)$ has a simple pole at $z=z_{0}$ and $(z-z_{0})^{2}g(z)$ is analytic at $z=z_{0}$ (the case $\lambda=-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm\sqrt{1+4\rho}$, where $\rho$ is the limiting value of $(z-z_{0})^{2}g(z)$ as $z\to z_{0}$. These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of $z_{0}$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
4: 21.10 Methods of Computation
• Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

• 5: Bonita V. Saunders
This work has resulted in several published papers presented as contributed or invited talks at universities and regional, national, and international conferences. …
6: 5.10 Continued Fractions
5.10.1 $\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,$
7: 14.31 Other Applications
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
8: 33.23 Methods of Computation
Bardin et al. (1972) describes ten different methods for the calculation of $F_{\ell}$ and $G_{\ell}$, valid in different regions of the ($\eta,\rho$)-plane. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_{0}$ and $G_{0}$ in the region inside the turning point: $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$.
9: 19.7 Connection Formulas
The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). … The second relation maps each hyperbolic region onto itself and each circular region onto the other: … The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …
10: 19.21 Connection Formulas
Change-of-parameter relations can be used to shift the parameter $p$ of $R_{J}$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). … For each value of $p$, permutation of $x,y,z$ produces three values of $q$, one of which lies in the same region as $p$ and two lie in the other region of the same type. In (19.21.12), if $x$ is the largest (smallest) of $x,y$, and $z$, then $p$ and $q$ lie in the same region if it is circular (hyperbolic); otherwise $p$ and $q$ lie in different regions, both circular or both hyperbolic. …